Chapter 8
Inference and Resolution for
Problem Solving
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Chapter 8 Contents (1)
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Normal Forms
Converting to CNF
Clauses
The Resolution Rule
Resolution Refutation
Proof by Refutation
Example: Graph Coloring
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Chapter 8 Contents (2)
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Normal Forms in Predicate Calculus
Skolemization
Unification
The Unification algorithm
The Resolution Algorithm
Horn Clauses in PROLOG
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Normal Forms
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A wff is in Conjunctive Normal Form (CNF)
if it is a conjunction of disjunctions:
 A1 Λ A2 Λ A3 Λ … Λ An
 where each clause, Ai, is of the form:
 B1 v B2 v B3 v … v Bn
 The B’s are literals.
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E.g.: A Λ (B v C) Λ (¬A v ¬B v ¬C v D)
Similarly, a wff is in Disjunctive Normal
Form (DNF) if it is a disjunction of
conjunctions.
E.g.: A v (B Λ C) v (¬A Λ ¬B Λ ¬C Λ D)
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Converting to CNF
Any wff can be converted to CNF by using
the following equivalences:
(1) A ↔ B  (A → B) Λ (B → A)
(2) A → B  ¬A v B
(3) ¬(A Λ B)  ¬A v ¬B
(4) ¬(A v B)  ¬A Λ ¬B
(5) ¬¬A  A
(6) A v (B Λ C)  (A v B) Λ (A v C)
 Importantly, this can be converted into an
algorithm – this will be useful when when we
come to automating resolution.
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Clauses
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Having converted a wff to CNF, it is usual to write
it as a set of clauses.
E.g.:
(A → B) → C
In CNF is:
(A V C) Λ (¬B V C)
In clause form, we write:
{(A, C), (¬B, C)}
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The Resolution Rule
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The resolution rule is written as follows:
A v B
¬ B v C
A v C
This tells us that if we have two clauses
that have a literal and its negation, we can
combine them by removing that literal.
E.g.: if we have {(A, C), (¬A, D)}
We would apply resolution to get {C, D}
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Resolution Refutation
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Let us resolve: {(¬A, B), (¬A, ¬B, C), A, ¬C}
We begin by resolving the first clause with the second
clause, thus eliminating B and ¬B:
{(¬A, C), A, ¬ C}
{C, ¬C}
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Now we can resolve both remaining literals, which
gives falsum:
^
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If we reach falsum, we have proved that our initial set
of clauses were inconsistent.
This is written:
{(¬A, B), (¬A, ¬B, C), A, ¬C} ╞ ^
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Proof by Refutation
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If we want to prove that a logical argument is
valid, we negate its conclusion, convert it to
clause form, and then try to derive falsum
using resolution.
If we derive falsum, then our clauses were
inconsistent, meaning the original argument
was valid, since we negated its conclusion.
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Proof by Refutation - Example
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Our argument is:
(A Λ ¬B) → C
A Λ ¬ B
C
Negate the conclusion and convert to clauses:
{(¬A, B, C), A, ¬B, ¬C}
Now resolve:
{(B, C), ¬B, ¬C}
{C, ¬C}
^
We have reached falsum, so our original argument
was valid.
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Refutation Proofs in Tree Form
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It is often sensible to represent a refutation proof in
tree form:
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In this case, the
proof has failed, as we
are left with E instead of falsum.
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Example: Graph Coloring
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Resolution refutation can be used to
determine if a solution exists for a
particular combinatorial problem.
For example, for graph coloring, we
represent the assignment of colors to the
nodes and the constraints regarding edges
as propositions, and attempt to prove that
the complete set of clauses is consistent.
This does not tell us how to color the
graph, simply that it is possible.
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Example: Graph Coloring (2)
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Available colors are r (red), g (green) and b (blue).
Ar means that node A has been coloured red.
Each node must have exactly one color:
Ar v Ag v Ab
¬Ar v ¬Ag
( Ar → ¬Ag)
¬Ag v ¬Ab
¬Ab v ¬Ar
If (A,B) is an edge, then:
¬Ar v ¬Br
¬Ab v ¬Bb
¬Ag v ¬Bg
Now we construct the complete set of clauses for our graph, and
try to see if they are consistent, using resolution.
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Normal Forms in Predicate Calculus
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A FOPC expression can be put in prenex normal form
by converting it to CNF, with the quantifiers moved to
the front.
For example, the wff:
(x)(A(x) → B(x)) → (y)(A(y) Λ B(y))
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Converts to prenex normal form as:
(x)(y)(((A(x) v A(y)) Λ (¬B(x) v A(y)) Λ
(A(x) v B(y)) Λ (¬B(x) v B(y))))
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Conversion to Prenex detail
(x)(A(x)→B(x))→(y)(A(y)ΛB(y))
 ¬(x)(A(x)→B(x))v(y)(A(y)ΛB(y))
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¬(x)(¬A(x)vB(x))v(y)(A(y)ΛB(y))
(x)¬(¬A(x)vB(x))v(y)(A(y)ΛB(y))
(x)(¬¬A(x)Λ¬B(x))v(y)(A(y)ΛB(y))
(x)(A(x)Λ¬B(x))v(y)(A(y)ΛB(y))
(x)(y)(A(x)Λ¬B(x))v(A(y)ΛB(y)))
(x)(y)(((A(x)vA(y))Λ(¬B(x)vA(y))Λ(A(x)
vB(y))Λ(¬B(x)v B(y))))
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Skolemization
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Before resolution can be applied,  must be
removed, using skolemization.
Variables that are existentially quantified are
replaced by a constant:
(x) P(x)
is converted to:
P(c)
c must not already exist in the expression.
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Skolem functions
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If the existentially quantified variable is within the
scope of a universally quantified variable, it must be
replaced by a skolem function, which is a function of
the universally quantified variable.
Sounds complicated, but is actually simple:
(x)(y)(P(x,y))
Is Skolemized to give:
(x)(P(x,f(x))
After skolemization,  is dropped, and the expression
converted to clauses in the usual manner.
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Unification (1)
To resolve clauses we often need to make
substitutions. For example:
{(P(w,x)), (¬P(y,z))}
 To resolve, we need to substitute y for w, and
z for x, giving:
{(P(y,z)), (¬P(y,z))}
 Now these resolve to give falsum.
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Unification (2)
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A substitution that enables us to resolve a set
of clauses is called a unifier.
We write the unifier we used above as:
{y/w, z/x}
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A unifier (u) is a most general unifier
(mgu) if any other unifier can be formed
by the composition of u with some other
unifier.
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An algorithm can be generated which obtains
a most general unifier for any set of clauses.
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The Resolution Algorithm
1 First, negate the conclusion and add it to the list
of assumptions.
2 Now convert the assumptions into Prenex Normal
Form
3 Next, skolemize the resulting expression
4 Now convert the expression into a set of clauses
5 Now resolve the clauses using suitable unifiers.
 This algorithm means we can write programs that
automatically prove theorems using resolution.
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Horn Clauses in PROLOG (1)
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PROLOG uses resolution.
A Horn clause has at most one positive
literal:
A v ¬B v ¬C v ¬D v ¬E …
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This can also be written as an implication:
BΛCΛDΛE→A
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In PROLOG, this is written:
A :- B, C, D, E
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Horn Clauses in PROLOG (2)
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Horn clauses can express rules:
A :- B, C, D
Or facts:
A :Or goals:
:- B, C, D, E
If a set of clauses is valid, PROLOG will
definitely prove it using resolution and
depth first search.
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Skolemization Example
(x)([a(X)Λb(X)]→[c(X,I)Λ(X)((Z)[c(X,Z)]→d(X,X))])v(X)(e(X))
(x)([a(X)Λb(X)]→[c(X,I)Λ(Y)((Z)[c(Y,Z)]→d(X,Y))])v(X)(e(X))
(x)(¬[a(X)Λb(X)]v[c(X,I)Λ(Y)((Z)[c(Y,Z)]→d(X,Y))])v(X)(e(X))
(x)(¬[a(X)Λb(X)]v[c(X,I)Λ(Y)((Z)[¬c(Y,Z)]vd(X,Y))])v(X)(e(X))
(x)([¬a(X)v¬b(X)]v[c(X,I)Λ(Y)((Z)[¬c(Y,Z)]vd(X,Y))])v(X)(e(X))
(x)([¬a(X)v¬b(X)]v[c(X,I)Λ(Y)((Z)[¬c(Y,Z)]vd(X,Y))])v(W)(e(W))
(x)(Y)(Z)(W)([¬a(X)v¬b(X)]v[c(X,I)Λ(¬c(Y,Z)vd(X,Y))]ve(W))
(x)(W)([¬a(X)v¬b(X)]v[c(X,I)Λ(¬c(f(X),g(X))vd(X,f(X)))])ve(W))
[¬a(X)v¬b(X)]v[c(X,I)Λ(¬c(f(X),g(X))vd(X,f(X)))])ve(W)
[¬a(X)v¬b(X)vc(X,I)ve(W)] Λ[¬a(X)v¬b(X)v¬c(f(X),g(X))vd(X,f(X))ve(W)
[¬a(X)v¬b(X)vc(X,I)ve(W)] Λ[¬a(U)v¬b(U)v¬c(f(U),g(U))vd(U,f(U))ve(V)
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